WANCO: Weak Adversarial Networks for Constrained Optimization problems

This paper focuses on integrating the networks and adversarial training into constrained optimization problems to develop a framework algorithm for constrained optimization problems. For such problems, we first transform them into minimax problems using the augmented Lagrangian method and then use two (or several) deep neural networks(DNNs) to represent the primal and dual variables respectively. The parameters in the neural networks are then trained by an adversarial process. The proposed architecture is relatively insensitive to the scale of values of different constraints when compared to penalty based deep learning methods. Through this type of training, the constraints are imposed better based on the augmented Lagrangian multipliers. Extensive examples for optimization problems with scalar constraints, nonlinear constraints, partial differential equation constraints, and inequality constraints are considered to show the capability and robustness of the proposed method, with applications ranging from Ginzburg--Landau energy minimization problems, partition problems, fluid-solid topology optimization, to obstacle problems.